Coordinate-wise Median: Not Bad, Not Bad, Pretty Good

We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in $\mathbb{R}^2$. For the minisum objective and an odd number of agents, we show that the coordinate-wise median mechanism (CM) has a worst-case approximation ratio (AR) of $\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}$. Further, we show that CM has the lowest AR for this objective in the class of anonymous and strategyproof mechanisms. For the $p-norm$ social welfare objective, we find that the AR for CM is bounded above by $2^{\frac{3}{2}-\frac{2}{p}}$ for $p\geq 2$. Since it follows from \citet{feigenbaum_approximately_2017} that any deterministic strategyproof mechanism must have AR at least $2^{1-\frac{1}{p}}$, our upper bound guarantees that the CM mechanism is very close to being the best deterministic strategyproof mechanism for $p\geq 2$. In particular, for $p=2$ (miniSOS) and $p=\infty$ (minimax), the AR of CM is in fact equal to the corresponding lower bound on any deterministic strategyproof mechanism of $\sqrt{2}$ and $2$ respectively. This leads us to conjecture that $AR(CM)=2^{1-\frac{1}{p}}$ for any $p \geq 2$.